Asymptotic Behavior of Solutions for a Free Boundary Problem with a Monostable Nonlinearity

Abstract

A free boundary problem with Dirichlet boundary conditions is studied. Such problems can be used for describing the spread of chemical substances or biological species, which live in a moving region $[0,h(t)].$ In this case, the free boundary $h(t)$ represents the spreading front. If the density of the substance or the population at the boundary exceeds a threshold value, they will be going to spread outwards. On the other hand, the outside environment may be not very beneficial for spreading and this generates a decay rate. We mainly analyze how the decay rate and threshold value affect the solutions. There is a trichotomy result: the solution is either shrinking, or the transition case, or spreading. Besides, if the decay rate or threshold value is large, only the shrinking happens.